I recommend people check out the great answer by elemtilas, but there is a point in the question that was overlooked there...
You are not terribly wealthy, and cannot afford many specialised or
high tech tools. Manufactured goods are vastly more expensive, so most
of your stuff is natural based and durable.
I would like to challenge this somewhat, but then to adhere to it after.
"Too long, didn't read" version
The essentials, and how to source them naturally, include...
cutting tools, probably made of stone. Break stones apart and use the sharp edges. Look for flint; it can help with fire starting and also can have a very sharp edge.
containers. You should try to stay near water as much as possible, but have containers to hold water for when you must leave the river. Make by burning out a hole in wood if you have nothing else.
There are no good natural items I know of that make good shelters which are light and small enough to easily bring with travelers. Trading for a tent or tarp will be high priority if possible.
Rope, make from plants or plant fibers.
Either flint & steel or friction fire tools (base board and either wood drill or wood plough) for starting fires.
If this is the same generation of people that lived through the apocalypse and are now facing the post-apocalypse, then there are likely to still be a lot of manufactured goods available. Not everything, but enough to get you some items.
In my country there are many times more knives than there are humans. There are lots of knives in stores. There are even more knives in the kitchens of almost every house in my culture; many of them wouldn't be the best survival knives, but they are better than nothing. Heck, my father used a knife sharpener on a butter knife and made a razor sharp camping knife out of it.
There might not be enough axes to go around, ditto for some other tools, but there will be enough that they are not super rare.
And everyone who has these things will find them so valuable fast that they would not sell them to the wealthy people unless they were starving and the wealthy provided the food they need.
And back to your restrictions...
The other answers provide reasonable lists, so I'll try to concentrate on their natural counterparts as per your restriction. I try not to go into too much detail or link to tutorials because this question is about a list of essential tools. If you want to go beyond that, Google the terms I use below.
Most post-apocalyptic stories I've heard of assume all the tools are scavenged from the remains of society. I like that you've suggested that is possible but not practical and are going for a more primitive, naturalistic feel. It is like jamming the stone age and the space age together and reconciling the two existing in tandem.
Almost everyone who gets into outdoor survival and gets even reasonably good at it says this same thing: if you learn some skills, your brain becomes more valuable than any physical tool. Using your knowledge, you can make tools or even get by without needing them.
A secondary cutting tool (ax, saw, hatchet, etc.) if you can manage it. Also these can substitute for many knife functions if you cannot get a knife.
If you manage to have iron/steel tools, a sharpener will keep them working better for a long time.
A good steel knife would be so useful to survivors in your scenario, especially to the traveling ones, that many people would literally kill for one.
You can get by with wood and stone cutting tools. They can get a job done, and you would do well to learn how to make and use them. But they wear out fast. This is not Minecraft - an iron axe doesn't just have three times the durability and work a little bit faster than a stone axe.
In reality, iron tools are vastly superior to their stone and wood counterparts. An iron tool properly cared for can last a lifetime, or even several lifetimes, and it will need to be sharpened a lot less often and will cut through things way, way faster than most natural alternatives. If you can manage in your scenario to get an iron/steel tool or two, especially a good knife, then you should go way out of your way to do so.
If you cannot get a good metal cutting tool, then you need to resort to sticks and stones to make cutting tools. You better get used to making them, as you'll need to keep making them many times for the rest of your life if you keep needing to cut through things. You can make stone cutting tools by breaking rocks apart to expose sharp edges; that is an overly simplified view of it, but that is the basic idea.
You can cut through wood with sharp rocks, and you can even cut through wood with sticks if you work at it long enough. To cut wood with wood, you literally just keep rubbing them together so that the two pieces of wood slowly wear each other away by friction. It takes a huge effort to do this, so what you need cut better be worth it or you might as well forget about it.
Food and water tools
A container(s) to carry water while away from water source
A way to boil the water if it could be biologically contaminated
There is a reason why many cities are near lakes or rivers. People used to settle near fresh water so they had all there water needs met. Do the same; travel between and along waterways. And use some containers to bring as much water with you as you can whenever you leave water behind.
If you don't have a manufactured container, you can make one.
Fold a leave into a cup.
Burn out a deep hole in a piece of wood to make it into a cup.
Scrape clean an animal hide, clean it, and fold it up into a cup.
Use anything you have to hold water even if that is not what it's meant for. Tarp, inside-out rain poncho, plastic bag, anything.
Learn to make clay containers.
If you think the water could be biologically contaminated (germs, viruses, or other bio), then boil it if you have something to boil water in. If you have a container that you do not want to put into a fire, then putting a hot rock from the fire into your container can boil the water.
For food, that is such a broad topic that you could ask multiple questions about food-gathering tools.
If you can get a tent or at least a tarp, great. But since we are concentrating on natural items...
If you are going to be traveling, then you really just need to keep setting up tiny shelters out of whatever you find along the way. You really just need to learn the art of making survival shelters. Anything you could make naturally would be either too big and heavy or take a very long time and a lot of skill to make.
Rope is so very useful. Unfortunately, it takes a lot of effort to create natural ropes in this setting. You need to figure out how to pull off the long, thin, flexible strands of fibers from plants and work them together to form rope.
In a pinch, if you just need a short length, lots of plant parts can be used as short ropes: some plant stems, the thin flexible ends of branches, vines, thin roots, and similar long, thin, tough plant parts.
Very, very good, dry fire-starting tinder, and any 1 of the following...
If you can manage to get a manufactured old item: a ferrocerium rod or a magnifying glass
Flint & steel
Friction fire base board and either wood drill or wood plough
Fires are not needed often, so maybe you could cook infrequently and borrow some hot coals from someone else's fire to start yours
Learn to find flint in the wild (it does not occur everywhere) and also learn to identify rocks with iron deposits on the surface and try to strike them together to make sparks. However, even if you have a good piece of flint and even if you strike it with an actual steel striker that was designed for striking flint, even then it is still not easy to get a fire started. Your travelers will need to do lots and lots (and lots and lots and...) of practice and have a mountain of patience.
If you have natural flint in your area, this is also great for another reason: flint can be used to make good cutting tools as well.
The other way your travelers might start fires is by a friction fire method. These methods are also hard. There are multiple different friction fire starting techniques, and I encourage you to look them up. The common names for these techniques (and each one of them is different) is: fire drill, bow drill, fire plough, fire saw, fire roll. Again, a lot of practice and patience is necessary.
Whatever method is used, you also need very good tinder. Thoroughly dried out grass, or certain plant seeds, or very thin wood shavings thoroughly dried, etc. - but whatever you use to light up, it needs to be very dry.
That covers most of the essentials, and as you can see you will need to learn how to make these things yourself or have something to trade with others who can make them.
So really, the number 1 thing you need is knowledge about how to make the things you need. Or, if you can get your hands on one in your scenario, a book that tells you what to do.
All of these naturally crafted things will keep wearing out fast. You will need to keep remaking them frequently. If you are in an area of cold, snowy winters, maybe you will spend your winter making tools for the coming year. If not, maybe you stop all other activity every once in a while to spend a week making tools.